DLMF:35.6.E2 (Q9803): Difference between revisions

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Property / constraint
 

ℜ ⁡ ( a ) > 1 2 ⁢ ( m - 1 ) 𝑎 1 2 𝑚 1 {\displaystyle{\displaystyle\Re(a)>\frac{1}{2}(m-1)}}

\realpart@@{(a)}>\frac{1}{2}(m-1)
Property / constraint: ℜ ⁡ ( a ) > 1 2 ⁢ ( m - 1 ) 𝑎 1 2 𝑚 1 {\displaystyle{\displaystyle\Re(a)>\frac{1}{2}(m-1)}} / rank
 
Normal rank
Property / constraint
 

ℜ ⁡ ( a ) > 1 2 ⁢ ( m - 1 ) 𝑎 1 2 𝑚 1 {\displaystyle{\displaystyle\Re(a)>\frac{1}{2}(m-1)}}

\Re(a)>\frac{1}{2}(m-1)
Property / constraint: ℜ ⁡ ( a ) > 1 2 ⁢ ( m - 1 ) 𝑎 1 2 𝑚 1 {\displaystyle{\displaystyle\Re(a)>\frac{1}{2}(m-1)}} / rank
 
Normal rank
Property / constraint
 

𝐓 ∈ 𝛀 𝐓 𝛀 {\displaystyle{\displaystyle\mathbf{T}\in{\boldsymbol{\Omega}}}}

\mathbf{T}\in{\boldsymbol{\Omega}}
Property / constraint: 𝐓 ∈ 𝛀 𝐓 𝛀 {\displaystyle{\displaystyle\mathbf{T}\in{\boldsymbol{\Omega}}}} / rank
 
Normal rank
Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
Defining formula:

d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1adec
Property / Symbols used
 
Property / Symbols used: Q10784 / rank
 
Normal rank
Property / Symbols used: Q10784 / qualifier
 
Defining formula:

∈ {\displaystyle{\displaystyle\in}}

\in
Property / Symbols used: Q10784 / qualifier
 
xml-id: introduction.Sx4.p1.t1.r10.m2adec
Property / Symbols used
 
Property / Symbols used: Q12718 / rank
 
Normal rank
Property / Symbols used: Q12718 / qualifier
 
Defining formula:

etr ⁡ ( 𝐗 ) exponential-trace 𝐗 {\displaystyle{\displaystyle\mathrm{etr}\left(\NVar{\mathbf{X}}\right)}}

\exptrace@{\NVar{\mathbf{X}}}
Property / Symbols used: Q12718 / qualifier
 
xml-id: C35.S1.p3.t1.r10.m3adec
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
Defining formula:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3adec
Property / Symbols used
 
Property / Symbols used: Q12725 / rank
 
Normal rank
Property / Symbols used: Q12725 / qualifier
 
Defining formula:

Γ m ⁡ ( a ) multivariate-Euler-Gamma 𝑚 𝑎 {\displaystyle{\displaystyle\Gamma_{\NVar{m}}\left(\NVar{a}\right)}}

\multivarEulerGamma{\NVar{m}}@{\NVar{a}}
Property / Symbols used: Q12725 / qualifier
 
xml-id: C35.S3.SS1.m1adec
Property / Symbols used
 
Property / Symbols used: Q10811 / rank
 
Normal rank
Property / Symbols used: Q10811 / qualifier
 
Defining formula:

ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}

\realpart@@
Property / Symbols used: Q10811 / qualifier
 
xml-id: C1.S9.E2.m1adec
Property / Symbols used
 
Property / Symbols used: Q12726 / rank
 
Normal rank
Property / Symbols used: Q12726 / qualifier
 
Defining formula:

a 𝑎 {\displaystyle{\displaystyle a}}

a
Property / Symbols used: Q12726 / qualifier
 
xml-id: C35.S1.XMD1.m1adec
Property / Symbols used
 
Property / Symbols used: Q12724 / rank
 
Normal rank
Property / Symbols used: Q12724 / qualifier
 
Defining formula:

𝐓 𝐓 {\displaystyle{\displaystyle\mathbf{T}}}

\mathbf{T}
Property / Symbols used: Q12724 / qualifier
 
xml-id: C35.S1.XMD11.m1adec
Property / Symbols used
 
Property / Symbols used: Q12719 / rank
 
Normal rank
Property / Symbols used: Q12719 / qualifier
 
Defining formula:

𝐗 𝐗 {\displaystyle{\displaystyle\mathbf{X}}}

\mathbf{X}
Property / Symbols used: Q12719 / qualifier
 
xml-id: C35.S1.XMD12.m1dec
Property / Symbols used
 
Property / Symbols used: Q12720 / rank
 
Normal rank
Property / Symbols used: Q12720 / qualifier
 
Defining formula:

𝛀 𝛀 {\displaystyle{\displaystyle{\boldsymbol{\Omega}}}}

{\boldsymbol{\Omega}}
Property / Symbols used: Q12720 / qualifier
 
xml-id: C35.S1.XMD13.m1dec
Property / Symbols used
 
Property / Symbols used: Q12728 / rank
 
Normal rank
Property / Symbols used: Q12728 / qualifier
 
Defining formula:

b 𝑏 {\displaystyle{\displaystyle b}}

b
Property / Symbols used: Q12728 / qualifier
 
xml-id: C35.S1.XMD2.m1adec
Property / Symbols used
 
Property / Symbols used: Q12723 / rank
 
Normal rank
Property / Symbols used: Q12723 / qualifier
 
Defining formula:

m 𝑚 {\displaystyle{\displaystyle m}}

m
Property / Symbols used: Q12723 / qualifier
 
xml-id: C35.S1.XMD8.m1dec

Latest revision as of 14:54, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:35.6.E2
No description defined

    Statements

    Defining formula
    Ψ ⁡ ( a ; b ; 𝐓 ) = 1 Γ m ⁡ ( a ) ⁢ ∫ 𝛀 etr ⁡ ( - 𝐓𝐗 ) ⁢ | 𝐗 | a - 1 2 ⁢ ( m + 1 ) ⁢ | 𝐈 + 𝐗 | b - a - 1 2 ⁢ ( m + 1 ) ⁢ d 𝐗 , confluent-hypergeometric-of-matrix-Psi 𝑎 𝑏 𝐓 1 multivariate-Euler-Gamma 𝑚 𝑎 subscript 𝛀 exponential-trace 𝐓𝐗 superscript 𝐗 𝑎 1 2 𝑚 1 superscript 𝐈 𝐗 𝑏 𝑎 1 2 𝑚 1 𝐗 {\displaystyle{\displaystyle\Psi\left(a;b;\mathbf{T}\right)=\frac{1}{\Gamma_{m% }\left(a\right)}\int_{\boldsymbol{\Omega}}\mathrm{etr}\left(-\mathbf{T}\mathbf% {X}\right)|\mathbf{X}|^{a-\frac{1}{2}(m+1)}\*{|\mathbf{I}+\mathbf{X}|}^{b-a-% \frac{1}{2}(m+1)}\mathrm{d}\mathbf{X},}}
    0 references
    DLMF id
    DLMF:35.6.E2
    0 references
    constraint
    ℜ ⁡ ( a ) > 1 2 ⁢ ( m - 1 ) 𝑎 1 2 𝑚 1 {\displaystyle{\displaystyle\Re(a)>\frac{1}{2}(m-1)}}
    0 references
    ℜ ⁡ ( a ) > 1 2 ⁢ ( m - 1 ) 𝑎 1 2 𝑚 1 {\displaystyle{\displaystyle\Re(a)>\frac{1}{2}(m-1)}}
    0 references
    𝐓 ∈ 𝛀 𝐓 𝛀 {\displaystyle{\displaystyle\mathbf{T}\in{\boldsymbol{\Omega}}}}
    0 references
    Symbols used
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
    0 references
    Q10784
    Defining formula
    ∈ {\displaystyle{\displaystyle\in}}
    xml-id
    introduction.Sx4.p1.t1.r10.m2adec
    0 references
    Q12718
    Defining formula
    etr ⁡ ( 𝐗 ) exponential-trace 𝐗 {\displaystyle{\displaystyle\mathrm{etr}\left(\NVar{\mathbf{X}}\right)}}
    xml-id
    C35.S1.p3.t1.r10.m3adec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
    0 references
    Q12725
    Defining formula
    Γ m ⁡ ( a ) multivariate-Euler-Gamma 𝑚 𝑎 {\displaystyle{\displaystyle\Gamma_{\NVar{m}}\left(\NVar{a}\right)}}
    xml-id
    C35.S3.SS1.m1adec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1adec
    0 references
    Q12726
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C35.S1.XMD1.m1adec
    0 references
    Q12724
    Defining formula
    𝐓 𝐓 {\displaystyle{\displaystyle\mathbf{T}}}
    xml-id
    C35.S1.XMD11.m1adec
    0 references
    Q12719
    Defining formula
    𝐗 𝐗 {\displaystyle{\displaystyle\mathbf{X}}}
    xml-id
    C35.S1.XMD12.m1dec
    0 references
    Q12720
    Defining formula
    𝛀 𝛀 {\displaystyle{\displaystyle{\boldsymbol{\Omega}}}}
    xml-id
    C35.S1.XMD13.m1dec
    0 references
    Q12728
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C35.S1.XMD2.m1adec
    0 references
    Q12723
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C35.S1.XMD8.m1dec
    0 references
     
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